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We give a simple characterization of which functions can be computeddeterministically by anonymous processes in dynamic networks, depending on thenumber of leaders in the network. In addition, we provide efficient distributedalgorithms for computing all such functions assuming minimal or no knowledgeabout the network. Each of our algorithms comes in two versions: one thatterminates with the correct output and a faster one that stabilizes on thecorrect output without explicit termination. Notably, these are the firstdeterministic algorithms whose running times scale linearly with both thenumber of processes and a parameter of the network which we call "dynamicdisconnectivity" (meaning that our dynamic networks do not necessarily have tobe connected at all times). We also provide matching lower bounds, showing thatall our algorithms are asymptotically optimal for any fixed number of leaders.  While most of the existing literature on anonymous dynamic networks relies onclassical mass-distribution techniques, our work makes use of a recentlyintroduced combinatorial structure called "history tree", also developing itstheory in new directions. Among other contributions, our results makedefinitive progress on two popular fundamental problems for anonymous dynamicnetworks: leaderless Average Consensus (i.e., computing the mean value of inputnumbers distributed among the processes) and multi-leader Counting (i.e.,determining the exact number of processes in the network). In fact, ourapproach unifies and improves upon several independent lines of research onanonymous networks, including Nedic et al., IEEE Trans. Automat. Contr. 2009;Olshevsky, SIAM J. Control Optim. 2017; Kowalski-Mosteiro, ICALP 2019, SPAA2021; Di Luna-Viglietta, FOCS 2022.

Optimal Computation in Leaderless and Multi-Leader Disconnected Anonymous Dynamic Networks